Type IIb supernovae (SNe) are important candidates to understand mechanisms that drive the stripping of stripped-envelope (SE) supernova (SN) progenitors. While binary interactions and their high incidence are generally cited to favor them as SN IIb progenitors, this idea has not been tested using models covering a broad parameter space. In this paper, we use non-rotating single- and binary-star models at solar and low metallicities spanning a wide parameter space in primary mass, mass ratio, orbital period, and mass transfer efficiencies. We find that our single- and binary-star models contribute to roughly equal, however small, numbers of SNe IIb at solar metallicity. Binaries only dominate as progenitors at low metallicity. We also find that our models can account for less than half of the observationally inferred rate for SNe IIb at solar metallicity, with computed rates 4% of core-collapse (CC) SNe. On the other hand, our models can account for the rates currently indicated by observations at low metallicity, with computed rates as high as 15% of CC SNe. However, this requires low mass transfer efficiencies ( 0.1) to prevent most progenitors from entering contact. We suggest that the stellar wind mass-loss rates at solar metallicity used in our models are too high. Lower mass-loss rates would widen the parameter space for binary SNe IIb at solar metallicity by allowing stars that initiate mass transfer earlier in their evolution to reach CC without getting fully stripped.